1. Introduction

Recently, there has been growing interest in designing and fabricating micro planar focusing optical devices due to its properties of small size, light weight and easy for integration. Various planar focusing devices have been reported [1–10], however, their spot size is hard to break the diffraction limit. A dipole-wave-reciprocal binary zone plate [11] was also ultilized to achieve sub-diffraction focusing with a very short focal length, less than one wavelength. Optical super-oscillation [12] is a new solution to break the diffraction limit and achieve far-field tight focus of light and super-resolution imaging [13]. Super-oscillatory micro lenses were demonstrated theoretically and experimentally [14–19], and in most cases, the lenses were based on binary amplitude mask. Due to the large sidelobes, present super-oscillatory lenses face the challenge of reducing the focus size while keeping sidelobes small. A possible way to suppress the sidelobe is to increase the numerical aperture of the lens [18], but this seems to suggest an inevitable large size in realizing a long focal length, which is favored in many practical applications. We also proposed and demonstrated a super-oscillation planar lens based on quasi-continuous amplitude modulation with width-varied sub-wavelength metallic slit array [20], which achieved super-oscillatory focusing with small sidelobes. According to the numerical study [21], for far-field super-oscillatory focusing, phase plays a very important role in improving the lens focusing performance and suppressing sidelboes. Binary phase filter [22] has been proposed to realize sub-diffraction focusing in combination with a high-numerical-aperture lens. A theoretical design of binary phase mask based super-oscillatory lens was also reported [23], however, the field of view of the lens was only 15 wavelengths, because of the difficulty in finding the best solution by using the optimization-free method. Here, we proposed and experimentally demonstrated a far-field super-oscillatory line-focusing of linearly polarized light by lens based on binary amplitude-phase mask with small sidelobes.

2. Theoretical consideration

Figure 1(a) depicts the focusing of transverse electric mode (TE) plane wave by a line-focusing lens based on binary amplitude-phase mask, and Fig. 1(b) shows the geometry of the L × W large lens. As shown in the figure, the basic unit of the lens is a stripe structure grown on the glass substrate. These unit structures are parallel to each other. The stripe generally consists of a Si₃N₄ layer and an aluminum layer. The size of a unit in y-direction is T, which is smaller than the working wavelength λ. The binary phase is achieved by the Si₃N₄ stripe thickness, 0 and t for phase change 0 and π, respectively. The binary amplitude transmittance of 0 and 1 was realized by a unit with and without a 100-nm thick aluminum film. In this way, amplitude and phase (A_i,φ_i) of ith unit can have three different combinations of (0, 0), (1, 0) and (1, π) for unit parameters (t_Al, t_Si3N4) of (100 nm, t), (0 nm, 0 nm) and (0 nm, t), respectively. The thickness of the Si₃N₄ layer is determined by t = λ/2(n _Si3N4-1), n _Si3N4 being the refractive index of Si₃N₄.

 

Fig. 1 (a) Far-field line-focusing of a plane wave with TE polarization by a micro lens based on binary amplitude-phase mask, and (b) the structure of the micro lens.

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For normal incident TE polarized plane wave at wavelength λ = 632.8 nm, a lens based on binary amplitude-phase mask was designed with a genetic algorithm [24] and angular spectrum method [25] for a given lens length L = 800λ, focal length f = 150λ, and focal line full width at half maximum (FWHM) 0.39λ. The numerical aperture (NA) of the lens is 0.936, and the corresponding diffraction limit is 0.534λ. The unit structure size of T is 500 nm, which is less than the working wavelength of 632.8 nm. The optimized lens amplitude and phase distributions are plotted in Fig. 2, which shows a clear variation in the spatial distribution of amplitude transmittance and phase delay on the lens output surface. COMSOL Multiphysics was used to conduct the numerical simulation of such lens. In the simulation, the thickness of the Si₃N₄ stripe was 298 nm, and the refractive index of the Si₃N₄ was 1.90. The focal plane was found to be 150.24λ away from the lens output surface. According to the simulation result, the normalized optical intensity distribution on the focal plane is as illustrated in Fig. 3. The inset of Fig. 3 shows the zoom-in plot of the intensity distribution. The focal line FWHM of the COMSOL simulation is 0.392λ, slightly larger than the theoretically designed FWHM of 0.39λ. The ratio of maximum sidelobe intensity to the central lobe intensity R_smax is 23%. In Fig. 3, it was noticed that, in the whole simulation area of [-1024λ, + 1024λ], no large sidelobe was observed. To ensure that there is no other sidelobe outside of this area, the total transmitted energy was calculated and compared with the total energy in the area of [-1024λ, + 1024λ] on the focal plane. It was found that the two values were exactly the same, indicating that there has no large sidelobes on the focal plane.

 

Fig. 2 The (a) amplitude and (b) phase distribution on the lens mask.

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Fig. 3 The theoretical intensity distribution on the focal plane.

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3. Results and discussions

Following the theoretical design, a microlens based on binary amplitude-phase mask was fabricated. On a 500-µm thick Pyrex 7740 glass wafer, a 298-nm thick Si₃N₄ film was grown, on the top of which was a 100-nm thick aluminum film. Then, according to the theoretical design shown in Fig. 2, electron-beam writing and dry etching were used to form the Si₃N₄ and aluminum strips. The scanning electron microscope (SEM) images of the lens were taken with Nova NanoSEM 430 + EDS. Figure 4(a) shows the SEM image of the 506 × 200 µm² microlens. Due to the small field of view, the SEM image of the two ends of the lens was distorted. Figure 4(b) is a zoom-in SEM image of the microlens.

 

Fig. 4 The SEM images of (a) the microlens and (b) the zoom-in image of the lens central part, where the red dashed line indicates the lens central line.

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To demonstrate the sub-diffraction focusing, an experimental setup was built up based on a 100-nm optical fiber probe (CFN-100 of Nanonics Imaging Ltd.) and a 3-D piezo nano-translation stage (P-561.3CD of Physik Instrumente GmbH & Co.) with spatial resolution around 10 nm for each axis. The scan range of the nano-translation stage is 100 μm for each of the x, y and z axes. The illumination light source is a 10-mW He-Ne laser (632.8 nm) with linear polarization. The beam 1/e diameter was expanded into 2.7 mm to guarantee a uniform illumination of the whole lens area. The lens was located at the beam center, and the beam was normally incident on the lens from the substrate side. The beam polarization was parallel to Si₃N₄ strips on the lens. By controlling the 3-D nano-translation stage, the nano-fiber probe can scan to obtain the optical intensity in the x-y plane at different distances from the micro-lens surface in the z direction. The collected photons were detected by a single photon detector (SPCM50A/M of Thorlabs Inc.)

In the experiment, the focal plane was found to be approximately 94 µm from the lens output surface, which was close to the theoretical focal length of 95 µm. Figure 5 shows the optical intensity distribution obtained in a small area of [20 µm, 28 µm] at the central part of the focal plane. The FWHM of the central peak is about 257 nm, or 0.406λ, and the sidelobe intensity was found to be less than 22% of the central peak. Both the FWHM and the sidelobe ratio were similar to the COMSOL Multiphysics simulation results of 0.392λ and 23%, respectively. The experimental value of the focal line FWHM 257 nm (0.406λ) was already smaller than the lens diffraction limit of 338 nm (0.534λ). The 9 nm (0.014λ) difference in FWHM between the experimental and theoretical results was majorly attributed to the mechanic shift of the 100-nm fiber probe during the scan.

 

Fig. 5 The intensity distribution in the nearby region of the central peak on the focal plane.

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To investigate the focal line property in the z-axis, the optical intensity distribution was obtained at different values of z along the optical axis near the focal plane. Figure 6(a) gives the color map of the corresponding optical intensity distribution in the y-z plane. Figure 6(b) plots the peak intensity and full width at half maximum as functions of z. It was found that, the central peak intensity first increases with z, and then reaches its maximum value of 362 at z = 93.45 μm, corresponding to the focal length. As the distance continues increasing, the peak intensity drops. However, the FWHM changes with z in an inverted trend. First, it decreases with z, and achieves its minimum value of 257 nm at the focal plane. Between z = 93.4 μm and 94.05 μm, FWHM increases gradually from 275 nm to 328 nm. Note that in the area of [93.35 μm, 94.05 μm] in Z axis, the value of the focal line FWHM was between 257 nm and 328 nm, which was smaller than the diffraction limit of 338 nm (0.534λ).

 

Fig. 6 (a) The color map of the optical intensity on the y-z plane; (b) The central peak intensity and FWHM along the z-axis near the focal plane.

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As being pointed above, theoretically, the lens has no large sidelobes on the focal plane. However, in the experiment, a single scan range was limited to 100 μm. For better understanding in the sidelobe distribution on the focal plane, two large scans were conducted in the negative and positive directions along the y-axis on the focal plane. Figure 7 depicts the optical intensity obtained by the two scans in normalized scale. It is clear that in the measured area of [-83.5 μm, + 75.9 μm], corresponding to [-132λ, + 120λ], the maximum sidelobe intensity is less than 22% of the central peak intensity, which is consistent with the COMSOL Multiphysics simulation. The results indicate a wide field of view of this sub-diffraction far-field focusing lens.

 

Fig. 7 The intensity distribution in a large region of [-132λ, + 120λ] on the focal plane.

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According to the super-oscillation criteria, a lens is super-oscillatory, if its focal spot size is smaller than 0.38λ/NA. The NA value of the microlens is 0.936 and the corresponding super-oscillation criteria is 0.406λ, which is just equal to the measured focal spot size. One important feature of optical super-oscillation is that it happens in a certain area where the modulus of local spatial frequency $\nabla \phi (x,y,z)$ is larger than the spatial cut-off frequency n₀/λ, and n₀ being the phase and medium refractive index respectively. Figures 8(a) and 8(b) plot the amplitude and phase distribution on the lens focal plane respectively. It was noticed that there are several positions where phase varies sharply and the phase change is about π. Those phase reversed positions correspond to the amplitude minima points, as shown in the Fig. 8(a). Figure 8(c) depicts the local spatial frequency on the focal plane, where k’ and k₀ are local wavenumber and wavenumber in medium respectively. The largest local spatial frequency is found at the first valley of the central peak, and its value is about 12 times of the spatial cut-off frequency k₀. This indicates a clear evidence of super-oscillatory, which squeezes the focal spot below the diffraction limit.

 

Fig. 8 (a)The amplitude, (b) phase, and (c) local wavenumber distribution on the focal plane, where k’ and k₀ are local wavenumber and wavenumber in medium respectively.

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5. Conclusion

Sub-diffraction focusing planar lens is attractive photonic device for applications in optical microscope, beam focusing, micro-fabrication and so on. We proposed and experimentally demonstrated a super-oscillatory microlens based on binary amplitude-phase mask for line-focusing of a linearly polarized plane wave. Experimental results showed a 0.406λ FWHM of the focal line, which was below the diffraction limit 0.534λ (0.5λ/NA) and equal to the super-oscillation criteria 0.406λ (0.38λ/NA). Moreover, the largest sidelobe intensity was found to be less than 22% of the central peak intensity within a large range of [-132λ, + 120λ], resulting in a clear field of view. Such lens can be extended to other spectrum ranges.